
Journal for Geometry and Graphics 28 (2024), No. 1, 019027 Copyright by the authors licensed under CC BY SA 4.0 In and Ex Spheres of a Tetrahedron Hidefumi Katsuura San Jose State University, San Jose, U.S.A., San Jose, U.S.A. hidefumi.katsuura@sjsu.edu We prove that (1) a tetrahedron is isosceles if and only if the vertices of its twin tetrahedron are the excenters of the tetrahedron, (2) if a tetrahedron is orthocentric, and if the orthocenter is either the incenter, the centroid, or the circumcenter, then the tetrahedron is regular, (3) a tetrahedron is regular if and only if the four exspheres are tangent to the insphere, and (4) we prove an inequality relating the inradius, circumradius, and the distances between the incenter and the vertices of a tetrahedron. Keywords: Insphere, incenter, inradius, exsphere, excenter, exradius, twin tetrahedron, isosceles tetrahedron, regular tetrahedron, centroid, circumsphere, circumradius, circumcenter, orthocentric tetrahedron, orthocenter, Lagrange multipliers. MSC: 51M04. [ Fulltextpdf (381 KB)] 